# String theory on the Schrödinger pp-wave background

## Abstract

We study string theory on the pp-wave geometry obtained by taking the Penrose limit around a certain null geodesic of the non-supersymmetric Schrödinger background. We solve for the spectrum of bosonic excitations and find compelling agreement with the dispersion relation of the giant magnons in the Schrödinger background obtained previously in [47]. Inspired by the pp-wave spectrum we conjecture an exact in the t’Hooft coupling dispersion relation for the magnons in the original Schrödinger background. We show that the pp-wave background admits exactly 16 Killing spinors. We use the explicit form of the latter in order to derive the supersymmetry algebra of the background which explicitly depends on the deformation parameter. Its bosonic subalgebra is of the Newton-Hooke type.

## Keywords

AdS-CFT Correspondence Integrable Field Theories Bosonic Strings SpaceTime Symmetries## Notes

### **Open Access**

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## References

- [1]J.M. Maldacena,
*The large N limit of superconformal field theories and supergravity*,*Int. J. Theor. Phys.***38**(1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetzbMATHGoogle Scholar - [2]
- [3]J. Ambjørn, R.A. Janik and C. Kristjansen,
*Wrapping interactions and a new source of corrections to the spin-chain/string duality*,*Nucl. Phys.***B 736**(2006) 288 [hep-th/0510171] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [4]N. Gromov, V. Kazakov and P. Vieira,
*Exact spectrum of anomalous dimensions of planar N*= 4*supersymmetric Yang-Mills theory*,*Phys. Rev. Lett.***103**(2009) 131601 [arXiv:0901.3753] [INSPIRE]. - [5]N. Beisert et al.,
*Review of AdS/CFT integrability: an overview*,*Lett. Math. Phys.***99**(2012) 3 [arXiv:1012.3982] [INSPIRE]. - [6]G. Georgiou, V.L. Gili and R. Russo,
*Operator mixing and the AdS/CFT correspondence*,*JHEP***01**(2009) 082 [arXiv:0810.0499] [INSPIRE].MathSciNetCrossRefGoogle Scholar - [7]G. Georgiou, V.L. Gili and R. Russo,
*Operator mixing and three-point functions in N*= 4*SYM*,*JHEP***10**(2009) 009 [arXiv:0907.1567] [INSPIRE]. - [8]G. Georgiou, V. Gili and J. Plefka,
*The two-loop dilatation operator of N*= 4*super Yang-Mills theory in the*SO(6)*sector*,*JHEP***12**(2011) 075 [arXiv:1106.0724] [INSPIRE]. - [9]K. Okuyama and L.-S. Tseng,
*Three-point functions in N*= 4*SYM theory at one-loop*,*JHEP***08**(2004) 055 [hep-th/0404190] [INSPIRE]. - [10]R. Roiban and A. Volovich,
*Yang-Mills correlation functions from integrable spin chains*,*JHEP***09**(2004) 032 [hep-th/0407140] [INSPIRE].MathSciNetCrossRefGoogle Scholar - [11]L.F. Alday, J.R. David, E. Gava and K.S. Narain,
*Structure constants of planar N*= 4*Yang-Mills at one loop*,*JHEP***09**(2005) 070 [hep-th/0502186] [INSPIRE]. - [12]G. Georgiou, V. Gili, A. Grossardt and J. Plefka,
*Three-point functions in planar N*= 4*super Yang-Mills theory for scalar operators up to length five at the one-loop order*,*JHEP***04**(2012) 038 [arXiv:1201.0992] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [13]D.E. Berenstein, J.M. Maldacena and H.S. Nastase,
*Strings in flat space and pp waves from N*= 4*super-Yang-Mills*,*JHEP***04**(2002) 013[hep-th/0202021] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [14]M. Spradlin and A. Volovich,
*Superstring interactions in a pp wave background*,*Phys. Rev.***D 66**(2002) 086004 [hep-th/0204146] [INSPIRE]. - [15]A. Pankiewicz and B. Stefanski, Jr.,
*PP wave light cone superstring field theory*,*Nucl. Phys.***B 657**(2003) 79 [hep-th/0210246] [INSPIRE]. - [16]P. Di Vecchia, J.L. Petersen, M. Petrini, R. Russo and A. Tanzini,
*The three string vertex and the AdS/CFT duality in the PP wave limit*,*Class. Quant. Grav.***21**(2004) 2221 [hep-th/0304025] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [17]S. Dobashi and T. Yoneya,
*Resolving the holography in the plane-wave limit of AdS/CFT correspondence*,*Nucl. Phys.***B 711**(2005) 3 [hep-th/0406225] [INSPIRE]. - [18]S. Lee and R. Russo,
*Holographic cubic vertex in the pp-wave*,*Nucl. Phys.***B 705**(2005) 296 [hep-th/0409261] [INSPIRE]. - [19]G. Georgiou and G. Travaglini,
*Fermion BMN operators, the dilatation operator of N*= 4*SYM and pp wave string interactions*,*JHEP***04**(2004) 001 [hep-th/0403188] [INSPIRE]. - [20]G. Georgiou, V.V. Khoze and G. Travaglini,
*New tests of the pp wave correspondence*,*JHEP***10**(2003) 049 [hep-th/0306234] [INSPIRE]. - [21]G. Georgiou and V.V. Khoze,
*BMN operators with three scalar impurites and the vertex correlator duality in pp wave*,*JHEP***04**(2003) 015 [hep-th/0302064] [INSPIRE]. - [22]C.-S. Chu, V.V. Khoze and G. Travaglini,
*Three point functions in N*= 4*Yang-Mills theory and pp waves*,*JHEP***06**(2002) 011 [hep-th/0206005] [INSPIRE]. - [23]G. Georgiou and D. Zoakos,
*Entanglement entropy of the N*= 4*SYM spin chain*,*JHEP***06**(2016) 099 [arXiv:1603.05929] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [24]J. Escobedo, N. Gromov, A. Sever and P. Vieira,
*Tailoring three-point functions and integrability*,*JHEP***09**(2011) 028 [arXiv:1012.2475] [INSPIRE]. - [25]Y. Jiang, I. Kostov, F. Loebbert and D. Serban,
*Fixing the quantum three-point function*,*JHEP***04**(2014) 019 [arXiv:1401.0384] [INSPIRE]. - [26]B. Basso, S. Komatsu and P. Vieira,
*Structure constants and integrable bootstrap in planar N*= 4*SYM theory*, arXiv:1505.06745 [INSPIRE]. - [27]Y. Kazama, S. Komatsu and T. Nishimura,
*Classical integrability for three-point functions: cognate structure at weak and strong couplings*,*JHEP***10**(2016) 042 [*Erratum ibid.***02**(2018) 047] [arXiv:1603.03164] [INSPIRE]. - [28]Y. Kazama and S. Komatsu,
*Three-point functions in the*SU(2)*sector at strong coupling*,*JHEP***03**(2014) 052 [arXiv:1312.3727] [INSPIRE].ADSCrossRefGoogle Scholar - [29]Y. Kazama and S. Komatsu,
*On holographic three point functions for GKP strings from integrability*,*JHEP***01**(2012) 110 [*Erratum ibid.***06**(2012) 150] [arXiv:1110.3949] [INSPIRE]. - [30]K. Zarembo,
*Holographic three-point functions of semiclassical states*,*JHEP***09**(2010) 030 [arXiv:1008.1059] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [31]M.S. Costa, R. Monteiro, J.E. Santos and D. Zoakos,
*On three-point correlation functions in the gauge/gravity duality*,*JHEP***11**(2010) 141 [arXiv:1008.1070] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [32]R. Roiban and A.A. Tseytlin,
*On semiclassical computation of*3*-point functions of closed string vertex operators in AdS*_{5}×*S*^{5},*Phys. Rev.***D 82**(2010) 106011 [arXiv:1008.4921] [INSPIRE]. - [33]G. Georgiou,
*Two and three-point correlators of operators dual to folded string solutions at strong coupling*,*JHEP***02**(2011) 046 [arXiv:1011.5181] [INSPIRE]. - [34]G. Georgiou, SL(2)
*sector: weak/strong coupling agreement of three-point correlators*,*JHEP***09**(2011) 132 [arXiv:1107.1850] [INSPIRE]. - [35]Z. Bajnok and R.A. Janik,
*Classical limit of diagonal form factors and HHL correlators*,*JHEP***01**(2017) 063 [arXiv:1607.02830] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [36]J. Maldacena, D. Martelli and Y. Tachikawa,
*Comments on string theory backgrounds with non-relativistic conformal symmetry*,*JHEP***10**(2008) 072 [arXiv:0807.1100] [INSPIRE]. - [37]C.P. Herzog, M. Rangamani and S.F. Ross,
*Heating up Galilean holography*,*JHEP***11**(2008) 080 [arXiv:0807.1099] [INSPIRE]. - [38]A. Adams, K. Balasubramanian and J. McGreevy,
*Hot spacetimes for cold atoms*,*JHEP***11**(2008) 059 [arXiv:0807.1111] [INSPIRE].ADSCrossRefGoogle Scholar - [39]M. Alishahiha and O.J. Ganor,
*Twisted backgrounds, PP waves and nonlocal field theories*,*JHEP***03**(2003) 006 [hep-th/0301080] [INSPIRE]. - [40]N. Beisert and R. Roiban,
*Beauty and the twist: the Bethe ansatz for twisted N*= 4*SYM*,*JHEP***08**(2005) 039 [hep-th/0505187] [INSPIRE]. - [41]C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie,
*Twisted Bethe equations from a twisted S-matrix*,*JHEP***02**(2011) 027 [arXiv:1010.3229] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [42]T. Matsumoto and K. Yoshida,
*Schrödinger geometries arising from Yang-Baxter deformations*,*JHEP***04**(2015) 180 [arXiv:1502.00740] [INSPIRE].ADSCrossRefGoogle Scholar - [43]H. Kyono and K. Yoshida,
*Supercoset construction of Yang-Baxter deformed AdS*_{5}×*S*^{5}*backgrounds*,*PTEP***2016**(2016) 083B03 [arXiv:1605.02519] [INSPIRE]. - [44]S.J. van Tongeren,
*Yang-Baxter deformations, AdS/CFT and twist-noncommutative gauge theory*,*Nucl. Phys.***B 904**(2016) 148 [arXiv:1506.01023] [INSPIRE]. - [45]C.A. Fuertes and S. Moroz,
*Correlation functions in the non-relativistic AdS/CFT correspondence*,*Phys. Rev.***D 79**(2009) 106004 [arXiv:0903.1844] [INSPIRE]. - [46]A. Volovich and C. Wen,
*Correlation functions in non-relativistic holography*,*JHEP***05**(2009) 087 [arXiv:0903.2455] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [47]G. Georgiou and D. Zoakos,
*Giant magnons and spiky strings in the Schrödinger/dipole-deformed CFT correspondence*,*JHEP***02**(2018) 173 [arXiv:1712.03091] [INSPIRE].ADSCrossRefGoogle Scholar - [48]C. Ahn and P. Bozhilov,
*Giant magnon-like solution in Sch*_{5}×*S*^{5},*Phys. Rev.***D 98**(2018) 106005 [arXiv:1711.09252] [INSPIRE]. - [49]M. Guica, F. Levkovich-Maslyuk and K. Zarembo,
*Integrability in dipole-deformed N*= 4*super Yang-Mills*,*J. Phys.***A 50**(2017) 39 [arXiv:1706.07957] [INSPIRE]. - [50]H. Ouyang,
*Semiclassical spectrum for BMN string in Sch*_{5}×*S*^{5},*JHEP***12**(2017) 126 [arXiv:1709.06844] [INSPIRE].ADSCrossRefGoogle Scholar - [51]G. Georgiou and D. Zoakos,
*Holographic three-point correlators in the Schrödinger/dipole CFT correspondence*,*JHEP***09**(2018) 026 [arXiv:1806.08181] [INSPIRE]. - [52]H. Dimov, M. Radomirov, R.C. Rashkov and T. Vetsov,
*On pulsating strings in Schrödinger backgrounds*, arXiv:1903.07444 [INSPIRE]. - [53]T. Mateos,
*Marginal deformation of N*= 4*SYM and Penrose limits with continuum spectrum*,*JHEP***08**(2005) 026 [hep-th/0505243] [INSPIRE].ADSCrossRefGoogle Scholar - [54]S.D. Avramis, K. Sfetsos and D. Zoakos,
*Complex marginal deformations of D*3*-brane geometries, their Penrose limits and giant gravitons*,*Nucl. Phys.***B 787**(2007) 55 [arXiv:0704.2067] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [55]H. Bacry and J. Levy-Leblond,
*Possible kinematics*,*J. Math. Phys.***9**(1968) 1605 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [56]G.W. Gibbons and C.E. Patricot,
*Newton-Hooke space-times, Hpp waves and the cosmological constant*,*Class. Quant. Grav.***20**(2003) 5225 [hep-th/0308200] [INSPIRE].ADSCrossRefGoogle Scholar - [57]K.T. Grosvenor, J. Hartong, C. Keeler and N.A. Obers,
*Homogeneous nonrelativistic geometries as coset spaces*,*Class. Quant. Grav.***35**(2018) 175007 [arXiv:1712.03980] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [58]J.M. Figueroa-O’Farrill and G. Papadopoulos,
*Homogeneous fluxes, branes and a maximally supersymmetric solution of M-theory*,*JHEP***08**(2001) 036 [hep-th/0105308] [INSPIRE]. - [59]M. Blau, J.M. Figueroa-O’Farrill, C. Hull and G. Papadopoulos,
*A new maximally supersymmetric background of IIB superstring theory*,*JHEP***01**(2002) 047 [hep-th/0110242] [INSPIRE].MathSciNetCrossRefGoogle Scholar - [60]M. Cvetič, H. Lü and C.N. Pope,
*M theory pp waves, Penrose limits and supernumerary supersymmetries*,*Nucl. Phys.***B 644**(2002) 65 [hep-th/0203229] [INSPIRE]. - [61]D. Sadri and M.M. Sheikh-Jabbari,
*The plane wave/super Yang-Mills duality*,*Rev. Mod. Phys.***76**(2004) 853 [hep-th/0310119] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [62]M. Sakaguchi and K. Yoshida,
*Non-relativistic AdS branes and Newton-Hooke superalgebra*,*JHEP***10**(2006) 078 [hep-th/0605124] [INSPIRE].